Free Probability Theory and its Applications in Operator Algebras

【摘要】沈PI将应用Voiculescu？s的“非交换”（或“自由”）概率论以算子代数为主题，其中包括冯·诺伊曼代数和C*-代数。对算子代数的研究通常被看作是对“非交换测度理论”的研究。non-commutative几何?。在过去的七十年里，这已经成为一个丰富而肥沃的研究领域。非交换测度空间中的非交换概率论（或自由概率论）是由D. Voiculescu在20世纪80年代提出的。利用非交换概率论的工具解决了算子代数领域的几个重要问题。在这个项目中，PI希望进一步发展自由概率论；目的是探讨算子代数领域的其他问题。特别是，他将研究Voiculescu？一元C*-代数的s拓扑自由熵理论搜索bdf -可拓半群不是群的C*-代数；来计算Voiculescu？有限von Neumann代数的s自由熵维；并研究冯·诺伊曼代数的生成问题。引入算子代数理论是为了获得量子力学基础的更严格的数学公式。从概率论的观点来看，自由概率论在算子代数领域有着惊人的应用。本项目旨在开发自由概率论中针对区域算子代数问题的新工具。这些问题的解将为我们分类冯诺依曼代数和C*代数提供新的方法。该项目的意义在于，自由概率论和算子代数理论的新发展一直在数学和物理的几个领域，如统计学和量子力学中有着深刻的应用。建议活动的智力价值：该项目是首席研究员之前在自由概率论，bdf -可拓半群和von Neumann代数的生成器问题方面的工作的继续。拟议活动的更广泛影响：该项目将在新罕布什尔大学加强研究发展的更广泛影响的地点进行。新罕布什尔大学的算子代数和算子理论研究小组由几位著名的数学家组成。对研究生的暑期支持将帮助研究生学习课程。该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。

AbstractShenThe PI will apply Voiculescu?s "non-commutative" (or "free") probability theory to the subject of operator algebras, which includes von Neumann algebras and C*-algebras. The study of operator algebras is often viewed as the study of "non-commutative measure theory" or ?non-commutative geometry?. Over the last seventy years, this has turned out to be a rich and fertile field of study. Non-commutative probability theory (or free probability theory) in the context of non-commutative measure spaces was developed by D. Voiculescu in the 1980's. Several important problems in the area of operator algebras were solved by the tools from non-commutative probability theory. In this project, the PI wishes to further develop free probability theory; with a view to attacking other problems in the area of operator algebras. In particular, he will study Voiculescu?s topological free entropy theory for unital C*-algebras; to search C*-algebras whose BDF-extension semigroups are not groups; to compute Voiculescu?s free entropy dimension for more finite von Neumann algebras; and to study the generator problems for von Neumann algebras.The theory of operator algebras was introduced to obtain a more rigorous mathematical formulation of the basics of quantum mechanics. From a probabilistic point view, free probability theory has surprising applications in the area of operator algebras. This project aims to develop new tools in free probability theory aiming at problems in the area operator algebras. The solutions to these problems will provide us with new ways to classify von Neumann algebras and C* algebras. The significance of the project lies in the fact that new developments in the theory of free probability and operator algebras have always had profound applications to several fields in mathematics and physics such as statistics and quantum mechanics. Intellectual Merit of the Proposed Activities: The project is the continuation of principal investigator's prior work in free probability theory, in BDF-extension semigroups, and in generator problem of von Neumann algebras. Broader Impact of the Proposed Activities: The project will take place at a location that strengthens the broader impacts of research development in the University of New Hampshire. The research group of Operator Algebra and Operator Theory in the University of New Hampshire consists of several well-established mathematicians. Summer support for graduate students will help graduate students to learn the subjects.This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5.